Certainly! Let's solve the equation step-by-step:
Given the equation:
[tex]\[ -8x = 42x^2 - 6 \][/tex]
1. Rearrange the equation:
Move all terms to one side to set the equation to zero:
[tex]\[ 42x^2 - 8x - 6 = 0 \][/tex]
2. Identify coefficients:
Compare the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(42x^2 - 8x - 6 = 0\)[/tex]:
[tex]\[
a = 42, \quad b = -8, \quad c = -6
\][/tex]
3. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-8)^2 - 4(42)(-6)
\][/tex]
[tex]\[
= 64 + 1008
\][/tex]
[tex]\[
= 1072
\][/tex]
5. Find the square root of the discriminant:
[tex]\[
\sqrt{1072} = \sqrt{16 \times 67} = 4\sqrt{67}
\][/tex]
6. Substitute the values into the quadratic formula:
[tex]\[
x = \frac{-(-8) \pm 4\sqrt{67}}{2(42)}
\][/tex]
[tex]\[
x = \frac{8 \pm 4\sqrt{67}}{84}
\][/tex]
7. Simplify the expression:
[tex]\[
x = \frac{8}{84} \pm \frac{4\sqrt{67}}{84}
\][/tex]
[tex]\[
x = \frac{2}{21} \pm \frac{\sqrt{67}}{21}
\][/tex]
So, the solutions to the equation are:
[tex]\[
x = \frac{2}{21} + \frac{\sqrt{67}}{21} \quad \text{and} \quad x = \frac{2}{21} - \frac{\sqrt{67}}{21}
\][/tex]
Therefore, the solutions can be written as:
[tex]\[
x = -\frac{2}{21} + \frac{\sqrt{67}}{21} \quad \text{and} \quad x = -\frac{\sqrt{67}}{21} - \frac{2}{21}
\][/tex]
Thus, the final solutions are:
[tex]\[
\boxed{\left[ -\frac{2}{21} + \frac{\sqrt{67}}{21}, -\frac{\sqrt{67}}{21} - \frac{2}{21} \right]}
\][/tex]
